Study of the adiabatic passage in tripod atomic systems in terms of the Riemannian geometry of the Bloch sphere.

2025-05-02 0 0 964.56KB 22 页 10玖币

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Study of the adiabatic passage in tripod atomic

systems in terms of the Riemannian geometry of the

Bloch sphere.

Arturs Cinins

Institute of Atomic Physics and Spectroscopy, University of Latvia, Jelgavas str. 3,

LV-1004 Riga, Latvia

E-mail: arturs.cinins@lu.lv

Martins Bruvelis

King Abdullah University of Science and Technology (KAUST), Computer, Electrical

and Mathematical Sciences and Engineering Division (CEMSE), Thuwal 23955-6900,

Saudi Arabia

Nikolai N. Bezuglov

Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034,

Russia

Rzhanov Institute of Semiconductor Physics SB RAS, Novosibirsk 630090, Russia

May 2022

Abstract.

We present an analysis of the stimulated Raman adiabatic passage processes based

on the methods of diﬀerential geometry. The present work was inspired by an excellent

article by Bruce W. Shore et al. (R. G. Unanyan, B. W. Shore, and K. Bergmann

Phys. Rev. A 59, 2910 (1999)). We demonstrate how a purely geometric interpretation

of the adiabatic passage in quantum tripod systems as a Riemannian parallel transport

of the dark state vector along the Bloch sphere allows describing the evolution of the

system for a given sequence of Stokes, pump and control laser excitation pulses. In

combination with the Dykhne-Davis-Pechukas adiabaticity criterion and the minimax

principle for circles on a sphere, this approach allows obtaining the analytical form

of the optimal laser pulse sequences for a high ﬁdelity tripod fractional STIRAP.

In contrast to the conventional STIRAP in Λ-systems, the Gaussian approximations

of the optimal laser pulse sequences allow reaching the inﬁdelity of 10−7for the

adiabaticity parameter of 300 without noticeable oscillatory or other detrimental eﬀects

on population transfer accuracy.

1. Introduction

Stimulated Raman adiabatic passage (STIRAP) is a robust method for selective

population transfer between quantum states with many applications in modern physics,

arXiv:2210.14847v1 [physics.atom-ph] 26 Oct 2022

Adiabatic passage in tripod systems 2

chemistry, and information processing [1, 2]. STIRAP processes in tripod systems [3, 4]

with their generalization on N-pod quantum systems [4] is of particular interest for

quantum information because of the two (or N-1) orthogonal dark states that can form

a qubit (or so called qudit) [5].

In atomic systems containing ndegenerate adiabatic states |Dii(d-states) with

constant energy εdindependent of the slowly varying parameters <nof the system, the

adiabatic passage has a number of speciﬁc features. These features are mainly due to the

presence of irremovable transitions in the subspace Λnof d-states caused by the operator

of nonadiabatic coupling. The eﬃciency of the corresponding non-adiabatic mixing of

d-states does not depend on the temporal scales of time-dependent system parameters

and, as noted in the physical literature, is determined by the geometry of the parameter

space, more precisely, the topology of a closed curve <n(t) formed by the parameters

upon a complete adiabatic cycle [6]. At the same time, it was established that the

temporal dynamics of d-states are reduced to a group b

U<of unitary transformations

in the subspace Λnand that gauge ﬁelds are the appropriate tool for describing such

transformations [7]. To the best of our knowledge, the ﬁrst study of gauge structure for

tripod STIRAP was made in a paper by Bruce W. Shore et al. [3], where the operators

U<were shown to form the orthogonal rotation group SO(2) of the two-dimensional

dark states subspace Λ2with ﬁnal evolution given by the value of the so-called geometric

phase [6].

(a)

N-1

(b)

Figure 1: Energy levels in tripod (N-pod) systems under the rotating wave

approximation with (a) lasers P, S, Q (, . . .) coupling the ground sublevels 1,3,4

(, . . . , N) to the excited state 2. The parameter ∆ corresponds to the single-photon

detuning. (b) Reduction of the linkage diagram (a) to a single coupled bright state Br

and a set of decoupled dark states D1,2(,...,N−1). Dashed lines represent the additional

sublevels |ji, j > 4 for N-pod systems with N > 3.

In this paper, we consider a somewhat diﬀerent, more geometric approach to solving

the problems of adiabatic passage based on the methods of diﬀerential and Riemannian

geometries [8, 9]. We proceeded from the remark made in the book [10] on interpreting

gauge ﬁelds as the cause of the curvature of the so-called charge space, which for tripod

systems is a kind of analog of d-subspace Λ2. The corresponding linkage diagram is

shown in Fig. 1a along with the excitation scheme (Fig. 1b) reducing to two dark and

one bright state (see details in Section 2.1). In Section 2.2, we will demonstrate that the

normalized bright state |Brican be associated with a unit vector eR=R/|R|, where

Adiabatic passage in tripod systems 3

the vector R∼(ΩP,ΩS,ΩQ) (Rabi vector) belongs to the three-dimensional parameter

space <3of laser Rabi frequencies Ωi(i=S, P, Q). Simultaneously, the unit vectors

associated with dark states |Dilie in the two-dimensional plane ΛRtangent to the unit

sphere at point eR. Temporal evolution of Ωi(t) determines a path eR(t) (Rabi path) on

the surface of a unit sphere (analogous to the Bloch sphere), while temporal evolution of

the dark state-vector due to nonadiabatic coupling is a direct consequence of Riemannian

parallel transport [10] of the ”dark” tangent planes ΛRalong the Rabi path. Based

on a purely geometric approach, in Section 3 analytical expressions are obtained for

optimal sequences of laser pulses that satisfy the Dykhne-Davis-Pechukas adiabaticity

criterion |R(t)|=const [3, 11, 12] and implement a fractional STRAP where only a

controlled fraction of population transfer occurs. Section 4 presents a series of numerical

simulations of the tripod quantum dynamics in the case of the most characteristic

STIRAP processes with optimal lasers pulse trains. Importantly, their temporal proﬁles

allow for Gaussian approximations that are convenient for experimental implementation,

while providing good population transfer accuracy with inﬁdelity reaching 10−7for the

adiabaticity parameter of 300.

Since our work is intended for a physical audience and does not imply knowledge

of the fundamentals of diﬀerential geometry, we have provided the description and

elementary proofs of the necessary provisions to the Appendices. In Appendix A, the

problem of the dark state evolution is considered within two diﬀerent frameworks: non-

adiabatic coupling due to quantum-mechanical eﬀects and the Riemannian procedure

of parallel transport of tangent planes. It is shown that a quantitative analysis of the

adiabatic passage in both approaches results in identical analytical expression (A.13)

for the local rotations of two-dimensional dark subspaces Λ2. Appendix B is devoted to

ﬁnding a convenient expression (B.10) for the geometric factor βthat determines the

fraction of the initial state population transferred in the STIRAP process in terms of a

contour integral over a closed-loop in the parameter space <3. An interesting feature

of the resulting expression (B.10) for βis related to the Dirac vector potential (B.10)

included in it, generated by a unit magnetic charge [13]. The last Appendix C contains

the mathematical details on the derivation of the excited state probability amplitude

(39) along with formula (40) for the inﬁdelity parameter of fractional STRAP.

Noteworthy, in contrast to most works that study the problems of tripod systems

and specify an artiﬁcial basis for dark states, a natural dynamic set of dark basis

states emerges from our approach, associated with geometric properties of curves in the

parameter space of the laser Rabi frequencies. The geometric approach for determining

the evolution of degenerate dark states during adiabatic passage can also be generalized

to N-pod atomic systems. Therefore, the wording of some provisions is given for the

case N > 3 where appropriate.

Adiabatic passage in tripod systems 4

2. Notation, assumptions and remarks

Preparation of quantum objects into a proper initial state with subsequent transfer

to another predeﬁned state is among the fundamental tasks of quantum optics and

informatics. One approach to solve such problems is to directly apply fractional STIRAP

to the tripod (or N-pod) systems with the energy levels diagram depicted in Fig. 1.

Speciﬁcally, we are concerned with producing a coherent superposition of states |1iand

|3iwhen the system is initially (at t=t0) in state |1i:

ψ(t=t0) = |1i → ψ(t=tf) = cos β|1i+ cos βeiγ|3i.(1)

We are dealing with one excited quantum state |2iand Nstable components |ii

(i= 1,3, ...N) of the ground state, which are subject to interaction with Nlaser ﬁelds.

The pump (P) and Stokes (S) lasers drive the level population transfer, while the other

N−2 control lasers play an auxiliary role.

For a given sequence of laser pulses, an exact analysis of the transformation (1),

parameterized by the mixing angle βand the relative phase γ, requires solving the

Schr¨odinger equation, which under the rotating wave approximation (RWA) reads:

dtψ=1

¯hc

Hψ;c

H=ε2|2ih2|+b

V; (2)

V=¯h

j=1,3,..

Ωj(t)|2ihj|+h.c. (3)

The ﬁrst term in the operator c

H(2) determines the energy structure of the N-pod

system: the energy εgof all degenerate bare ground sublevels |jiis taken equal to zero,

while the energy ε2of the upper bare state |2icorresponds to the single-photon detuning

∆ of lasers: ε2= ¯h∆. The operator b

V(t) (3) describes the atomic levels coupling with

the laser ﬁelds via their slow varying Rabi frequencies Ωj= (2/¯h)h2|b

V|ji. According

to (1), the initial condition for the state vector ψimplies ψ(t=t0) = |1i.

The key point for our approach is the ability to assume, without a loss of generality,

that (i) Rabi frequencies Ωjof all lasers are real and (ii) the relative phase β(1) is equal

to zero (for the relevant rationale see [14] when discussing the formulas (4-7) given

there).

2.1. Bright and dark states in N-pod systems

The implementation of passage (1) without uncontrolled phase losses assumes the

absence of mixing, generated by the ﬁeld operator b

V(3), between the state-vector

ψand the unstable upper level 2. To avoid possible dephasing processes, one needs to

support the embedding of the vector ψ(t) in the subspaces ΛD(t) of dark states, which

become time-dependent when the Rabi frequencies alter. The criterion for the vector

|Di=X

j6=2

C(D)

j|ji(4)

Adiabatic passage in tripod systems 5

to belong to the category of dark states reduces to zeroing the matrix element h2|b

V|Di

that means [15]:

h2|b

V|Di=¯h

j6=2

ΩjC(D)

j= 0.(5)

Equations (4), (5) may be treated as an orthogonal condition

hBr|Di= 0; |Bri=X

j6=2

Ωj|ji/sX

j6=2

Ω2

j(6)

between any dark state (4) and the newly introduced unit wave vector |Bri(6).

Straightforward calculation yields

¯hh2|b

V|Bri=1

2sX

j6=2

Ω2

j≡1

2Ωeff ,(7)

i.e. the unit vector |Briappears, in contrast to all decoupled dark |Disubstates, to be

strongly coupled to the excited state 2, the linkage constant Ωef f having played the role

of eﬀective Rabi frequency. For this reason, the state |Brican be termed ”bright”.

Equation (6) implies that the subspace ΛD, composed from the dark states is

orthogonal to the one-dimensional subspace ΛBr, containing the single bright state |Bri.

The dimension of ΛD, thus, is N−1, i.e., one can choose N−1 mutually orthogonal dark

states |Dki. The corresponding linkage diagram for the the ﬁeld operator b

V(3) in the

basis |Bri,|Dkiis depicted in Fig. 1(b). Noteworthy, upon altering Rabi frequencies,

both subspaces ΛBr(t),ΛD(t) become time-dependent.

The N-pod operator c

H(2) acts independently in the subspace ΛDof dark states

and subspace Λ±of two coupled |Bri,|2ivectors. Diagonalization of c

Hin Λ±results

in the formation of two adiabatic (dressed) states |±i as superpositions of vectors |Bri

and |2i[14, 15] with repulsive adiabatic energies

ε±(t) = ¯h

2∆±¯h

2q∆2+ Ωeff (t)2(8)

that alter as the laser pulses pass. At the same time, the dark states |Diobey the

relation c

H|Di= 0, i.e. all |Diare degenerate adiabatic states with zero energy εD≡0,

regardless of laser coupling strengths.

Although the four-level system (N= 3), which we will focus on below, will be

suﬃcient to perform an operation (1), increasing the number of degrees of freedom

makes N-pod systems more ﬂexible and allows more complex quantum transformations

[5]. Importantly, the conﬁgurations of dark states, depicted in Fig. 1b are of the same

type so that the basic ideas of the geometric approach as applied to tripod STIRAP

also work in the general situation with N > 3.

2.2. Geometrical counterparts of Bright and Dark states

Since all the coeﬃcients in relations (4-7) are real, we can associate the set of Rabi

frequencies Ωjthat deﬁne the bright state wave function |Bri(6) as components of

the Rabi vector R∼(ΩP,ΩS,ΩQ, ..., ΩN) in some Euclidean parameter space <N. In

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StudyoftheadiabaticpassageintripodatomicsystemsintermsoftheRiemanniangeometryoftheBlochsphere.ArtursCininsInstituteofAtomicPhysicsandSpectroscopy,UniversityofLatvia,Jelgavasstr.3,LV-1004Riga,LatviaE-mail:arturs.cinins@lu.lvMartinsBruvelisKingAbdullahUniversityofScienceandTechnology(KAUST),Computer,E...

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Study of the adiabatic passage in tripod atomic systems in terms of the Riemannian geometry of the Bloch sphere.

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